GGA + U Study of the Optical Properties of LiH

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1. Introduction

The Alkali hydrides compounds XH (X = Li, Na, K, Rb and Cs) are very simulating compounds in that they have the simplest electronic band structure. Alkali hydrides with complex hydrides are besides interesting for their capacity to store hydrogen [1]. The electronic band structure, and structural property (lattice constant) of LiH, in rock salt structure, with space groups Fm3m, in cubic (FCC) system have been examined, using density functional theory, employing the full-potential linearized augmented plane wave (FP-LAPW) [2]. Generalized gradient approximation and GGA + U were used as exchange correlation potentials, with WIEN2k code; detailed method of calculation of the observables was explicated in [3]. To comprehend the optical properties of solids, their electronic band structure provides a more perspicuous explanation. [2] has studied the dielectric function of LiH in RS structure by first-principles density and all electronic GW approximation, with the latter known for its unconscionable computational cost. Confirming with [1], the optical properties for the family of alkali hydride compounds have been done for only the LiH and the NaH compounds, in RS structure, using GGA as the exchange correlation (XC).

In this present work, the optical properties of LiH are to be investigated, using full-potential linearized augmented plane wave (FP-LAPW), using GGA and GGA + U approximations with WIEN2k codes in the framework of density function theory (DFT) [4].

2. Theoretical Consideration

Dielectric Function

The optical properties can be derived from the dielectric function $\epsilon \left(\omega \right)$ computed by density functional theory (DFT). Dielectric function is a three-dimensional tensor, which hinges on the symmetry of the crystal, and is calculated directly from the Kohn-Sham energy eigenvalues, ${\epsilon}_{k}$. In the Random Phase Approximation (RPA), the function, ${\epsilon}_{ij}\left(\omega \right)$, can be expressed as [5]

$\begin{array}{c}{\epsilon}_{ij}={\delta}_{ij}-\frac{1}{v{\omega}^{2}}{{\displaystyle \sum}}_{n,k}\left(\frac{-\delta F\left(\epsilon \right)}{\delta {\epsilon}_{{\epsilon}_{n,k}}}\right){P}_{i;n,n,k}{P}_{j;n,n,k}\\ \text{\hspace{0.17em}}\text{\hspace{0.05em}}-\frac{4\pi}{v{\omega}_{2}}{\displaystyle \sum}\frac{{P}_{i;c,v,k}{P}_{j;c,v,k}}{\left({\epsilon}_{c,k}-{\epsilon}_{v,k}\right){\left({\epsilon}_{c,k}-{\epsilon}_{v,k}\right)}^{2}}\end{array}$ (1)

where V is a unit cell Volume, ${P}_{n\mathrm{,}m\mathrm{,}k}$ are momentum matrix elements between the bands n and m, for the point K of the crystal. $F\left(\epsilon \right)$ is a Fermi-Dirac distribution function:

$F\left(\epsilon \right)=\frac{1}{\mathrm{exp}\left(\frac{\epsilon -{\epsilon}_{F}}{{K}_{B}T}\right)+1}$ (2)

where ${K}_{B}$ is Boltzmann constant.

3. Optical Properties of LiH

3.1. Imaginary and Real Parts of the Dielectric Function

The imaginary part, ${\epsilon}_{2}\left(\omega \right)$, for the dielectric function $\epsilon \left(\omega \right)={\epsilon}_{1}\left(\omega \right)+i{\epsilon}_{2}\left(\omega \right)$, can be calculated using momentum matrix [6]. The corresponding eigenfunction of each of the occupied and unoccupied state contributes to the matrix elements. The real parts ${\epsilon}_{1}\left(\omega \right)$ of the dielectric function can be derived from the imaginary part ${\epsilon}_{2}\left(\omega \right)$ by Krong-Kramers relationship [6].

The imaginary part of the dielectric function is indicative of real transfer between occupied and unoccupied states, thus the imaginary part then handles the attenuation, while the real part accounts for refraction; explicitly, the real part indicates scattering and loss in optical processes. At this point, it is apt to maintain that the conduction band is the imaginary part, while the valence band is the real part of the dielectric function.

3.2. Refractive Index and Extinction Coefficient

The refractive index is one of the principles defining the characteristics of an optical material, while the extinction coefficient, K, illustrates the exponential decay of the amplitude of the electromagnetic waves. The refractive index and the extinction coefficients are intrinsically related, for they are derived from the same physical process. The refractive index and the extinction coefficient are tensors, and expressed as in Equations (3) and (4) below:

${n}_{ii}\left(\omega \right)=\sqrt{\frac{\left|{\epsilon}_{ii}\left(\omega \right)\right|+Re{\epsilon}_{ii}\left(\omega \right)}{2}}$ (3)

and

${k}_{ii}\left(\omega \right)=\sqrt{\frac{\left|{\epsilon}_{ii}\left(\omega \right)\right|-Re{\epsilon}_{ii}\left(\omega \right)}{2}}$ (4)

where ${n}_{ii}\left(\omega \right)$ is the refractive index, and ${k}_{ii}\left(\omega \right)$ is the extinction coefficient.

3.3. Reflectivity and Absorption Coefficient

In optical experiments, ${n}_{ii}\left(\omega \right)$ and ${k}_{ii}\left(\omega \right)$ cannot be measured explicitly. The observables are reflectivity ${R}_{ii}\left(\omega \right)$, and the absorption coefficient ${A}_{ii}\mathrm{(}\omega \mathrm{)}$. The absorption coefficient describes how the intensity of light decays. It can be shown in Literature on electromagnetism that these quantities can be expressed as in Equation (5):

${R}_{ii}\left(\omega \right)=\frac{{\left({n}_{ii}\left(\omega \right)-1\right)}^{2}+{k}_{ii}^{2}\left(\omega \right)}{{\left({n}_{ii}\left(\omega \right)+1\right)}^{2}+{k}_{ii}^{2}\left(\omega \right)}$ (5)

and

${A}_{ii}\left(\omega \right)=\frac{2\omega {k}_{ii}\left(\omega \right)}{c}$ (6)

4. Computational Methods

To determine the optical properties of LiH, a self-scheme was used by solving the Kohn-Sham equation, employing FP-LAPW method, in the frame work of density theory along with GGA and GGA + U functionals [4] by WIEN2K codes, details of which are discussed in electronic and structural properties, using GGA and GGA + U as exchange correlation potentials [3].

5. Results and Discussion

Absorption Edge for GGA and GGA + U Functional

The calculated complex dielectric function for LiH is depicted in Figure 1. The

Figure 1. Real and imaginary parts of the dielectric function of LiH.

familiar method in the GGA calculations of scissor operation, shifting the empty conduction bands (imaginary part of the dielectric function) upwards to correspond with the experimental scissor operation was observed [7]. It was recognized that the first absorption edge for GGA and GGA + U match together, albeit, the fine structure at higher energy slightly differ for the improved XC. It was, also, noticed that first absorption peak (transition at point X) was in good agreement with that calculated by [2], which was 9 eV (approximately). The discrepancies noticed at other peaks due to the application of GGA + U.

The imaginary part of the dielectric function obtained in this way depicts the absorption spectra of LiH. The black lines are for the GGA-PBE calculations, where the scissor operation was employed to correct the band gap error of this exchange-correlation functional [8].

6. Conclusion

The dielectric function of Lithium hydride (LiH), which is the underlying quantity relating to its electronic structure, and outlines its optical properties, has been calculated. It was observed that the application of the GGA + U approximation has very simple consequence for the simplicity of the band structure of alkali hydrides.

References

[1] Anderson, O.K. (1975) Linear Methods in Band Theory. Physical Review B, 12, 3060. https://doi.org/10.1103/PhysRevB.12.3060

[2] Setten, V., Popa, V.A. and de Wijs, G.A.B. (2007) Electronic Structure and Optical Properties of Lightweight Metal Hydrides. Physical Review B, 15, 35204.
https://doi.org/10.1103/PhysRevB.75.035204

[3] Uko, O., Michael, U.O. and Udoimuk, A.B. (2014) Electronic and Structural Properties of CaH2, Using GGA and GGA + U Approximations, with WIEN2k Codes. Innovative Space of Scientific Research Journals, 27, 252-262.

[4] Wu, H., Zhou, W., Udovic, T.J., Rush, J.J. and Yildirim, J.J. (2007) Structure and Vibrational Spectra of Calcium Hydroxide and Deuteride. Journal of Alloys and Compound, 436, 51-52. https://doi.org/10.1016/j.jallcom.2006.07.042

[5] Blaha, P., Schwarz, K. and Madsen, G.K. (2001) Electronic Calculations of Solid Using WIEN2K Package for Material Sciences. Computer Physics Communications, 147, 71-76. https://doi.org/10.1016/S0010-4655(02)00206-0

[6] Ambrosch-Daxl, C. and Sofo, J.O. (2006) Linear Properties of Solids within the Full-Potential Linearized Augmented Plane Wave Method. Physics, Institute for Computational and Data Sciences (ICDS), Material Research Institute (MRI).

[7] Starace, A.F. and Siamak, S. (1982) Random-Phase Approximation to One-Body Transition Metal Element for Open-Shell Atoms. Physics Review A, 25, 2135.
https://doi.org/10.1103/PhysRevA.25.2135

[8] El Gridani, A. and Mohammed, E.M. (2000) Electronic and Structural Properties of CaH2: An Nition Hatree-Fock Study. Chemical Physics, 252, 1-8.
https://doi.org/10.1016/S0301-0104(99)00333-X